Strategy-proofness and efficiency with non-quasi-linear preferences: A characterization of minimum price Walrasian rule

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1 Theoretical Economics 10 (2015), / Strategy-proofness and efficiency with non-quasi-linear preferences: A characterization of minimum price Walrasian rule Shuhei Morimoto Graduate School of Economics, Kobe University Shigehiro Serizawa Institute of Social and Economic Research, Osaka University We consider the problem of allocating objects to a group of agents and how much agents should pay. Each agent receives at most one object and has nonquasi-linear preferences. Non-quasi-linear preferences describe environments where payments influence agents abilities to utilize objects or derive benefits from them. The minimum price Walrasian (MPW) rule is the rule that assigns a minimum price Walrasian equilibrium allocation to each preference profile. We establish that the MPW rule is the unique rule satisfying strategy-proofness, efficiency, individual rationality, andno subsidy for losers. Since the outcome of the MPW rule coincides with that of the simultaneous ascending (SA) auction, our result supports SA auctions adopted by many governments. Keywords. Minimum price Walrasian equilibrium, simultaneous ascending auction, strategy-proofness, efficiency, heterogeneous objects, non-quasi-linear preferences. JEL classification. D44, D47, D71, D82. Shuhei Morimoto: morimoto@people.kobe-u.ac.jp Shigehiro Serizawa: serizawa@iser.osaka-u.ac.jp We are very grateful to the co-editor and four anonymous referees for their many detailed and helpful comments. This article was presented at the conference honoring Barberà at the Universitat Autònoma de Barcelona in 2011, Frontiers of Market Design at Ascona, Switzerland in 2012, the 2012 Annual Conference of the Association for Public Economic Theory, the 2012 Meeting of the Society for Social Choice and Welfare, the 2012 Autumn Meeting of the Japanese Economic Association, the 2013 North American Summer Meeting of the Econometric Society, the 2013 Conference on Economic Design, the 2013 Asian Meeting of the Econometric Society, and CIREQ Montreal Matching Conference in We thank participants at those conferences for their valuable comments. We are also grateful to seminar participants at Hitotsubashi, Indian Statistical Institute, Keio, Kobe, Kyoto, Rice, Rochester, Stanford, Tohoku, Tokyo, and Waseda for their comments. We specially thank Ahmet Alkan, Tommy Andersson, Itai Ashlagi, Salvador Barberà, Anna Bogomolnaia, Jeremy Bulow, Atsushi Kajii, Fuhito Kojima, Paul Milgrom, Debasis Mishra, Hervé Moulin, Shinji Ohseto, Motty Perry, Marek Pycia, John Roberts, Alvin Roth, Toyotaka Sakai, James Schummer, Ilya Segal, Arunava Sen, Lars-Gunnar Svensson, William Thomson, and Jun Wako for their detailed comments. Morimoto is a Research Fellow of Japan Society for the Promotion of Science (JSPS), and gratefully acknowledges the financial support from JSPS Research Fellowships for Young Scientists (JSPS KAKENHI Grant ). Remaining errors are ours. Copyright 2015 Shuhei Morimoto and Shigehiro Serizawa. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at DOI: /TE1470

2 446 Morimoto and Serizawa Theoretical Economics 10 (2015) 1. Introduction 1.1 Purpose Since the 1990s, governments in numerous countries have conducted auctions to allocate a variety of objects or assets including spectrum rights, vehicle ownership licenses, and land. Although auctions sometimes make a large amount of government revenue, the announced goals of many government auctions are rather to allocate objects efficiently, i.e., to agents who benefit most from them. 1 Agents who benefit more are willing to pay higher prices and thus, have a better chance to win the auctions. However, as mentioned below, large-scale auction payments would influence agents abilities to utilize objects or benefit from them, thereby complicating efficient allocations. This article analyzes rules that allocate auctioned objects efficiently even when payments are so large that they impair agents abilities to utilize them or realize their benefits. We investigate what types of allocation rules can allocate objects efficiently in such environments. 1.2 Main result An allocation rule, orsimplyarule, is a function that assigns to each preference profile an allocation, which consists of an assignment of objects and agents payments. Each agent receives one object at most, and has a preference over objects and payments. 2 The domain of rules is the class of preference profiles. We assume that preferences satisfy monotonicity, 3 continuity, and finiteness, which means that, given an assignment, any change of assigned object is compensated by a finite amount of money. We call such preferences classical. It is well known that in this model, there is a minimum price Walrasian equilibrium (MPWE), 4 and that the allocation associated with the MPWE coincides with the outcome of a certain type of auction called the simultaneous ascending (SA) auction. 5 Under SA auctions, bids on all objects start simultaneously, and the sale of any object is not settled as long as new bids are made on some objects. We focus on the rule that assigns an MPWE allocation to each preference profile. We refer to this rule as the minimum price Walrasian (MPW) rule. The MPW rule satisfies four desirable properties. The first is (Pareto) efficiency. An allocation is efficient if no agent can be made better off without either some other agents being made worse off or the government s revenue being reduced. 6 The second is strategy-proofness. Note that efficiency is evaluated based on agents preferences. Thus, an efficient allocation cannot be chosen without information about preferences. Since 1 For example, frequency auctions in the United States were introduced to promote efficient and intensive use of the electromagnetic spectrum. See McAfee and McMillan (1996, p. 160). 2 Each agent knows his own preference. In this sense, our model is one of private value models. 3 More precisely, in this article, we introduce two types of monotonicity assumptions, which we call money monotonicity and desirability of objects. See Section 2 for the formal definitions. 4 See Demange and Gale (1985). 5 For example, see Demange et al. (1986). 6 In our auction model, efficiency is defined by taking government revenue into account.

3 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 447 preferences are private information, agents may have an incentive to behave strategically to influence the final outcome in their favor. Strategy-proofness is an incentivecompatibility property, which gives a strong incentive for each agent to reveal his true preference. It says that for each preference profile, in the normal form game induced by the rule, it is a (weakly) dominant strategy for each agent to reveal his true preference. The MPW rule satisfies strategy-proofness 7 and chooses an efficient allocation corresponding to the revealed preferences. The third property of the MPW rule is individual rationality, which requires that no agent should be made worse off than if he had received no object and paid nothing. This property induces voluntary participation. The fourth property is no subsidy for losers. Under the MPW rule, the governments never subsidize losers. This property prevents agents who do not need objects from flocking to auctions only to sponge subsidies. The primary conclusion of this article is that only the minimum price Walrasian rule satisfies strategy-proofness, efficiency, individual rationality, and no subsidy for losers (Theorem 2). Since the outcome of the MPW rule coincides with that of the SA auction (Proposition 1), the result supports SA auctions adopted by many governments. 1.3 Related literature Holmström (1979) establishes a fundamental result relating to our question that applies when agents benefits from auctioned objects are not influenced by their payments, i.e., agents have quasi-linear preferences. He assumes that preferences are quasi-linear, and shows that only the Vickrey Clarke Groves (VCG) 8 type allocation rules satisfy strategy-proofness and efficiency. 9 His result implies that on the quasi-linear domain, only the Vickrey rule 10 satisfies strategy-proofness, efficiency, individual rationality, and no subsidy for losers. 11 As Marshall (1920) demonstrates, preferences are approximately quasi-linear if payments for goods we analyze are sufficiently low. 12 However, quasilinearity is not an appropriate assumption for large-scale auctions. Excessive payments for the auctioned objects may damage bidders budgets to purchase complements for effective uses of the objects and thus, may influence the benefits from the objects. Alternatively, bidders may need to obtain loans to bid high amounts, and typically financial costs are nonlinear in borrowings, which makes bidders preferences on objects and payments non-quasi-linear. 13 In spectrum license auctions and vehicle ownership 7 In addition, the MPW rule is group strategy-proof, i.e., by jointly misrepresenting their preferences, no group of agents should obtain assignments that they prefer. 8 See Vickrey (1961), Clarke (1971), and Groves (1973). 9 More precisely, Holmström (1979) studies public goods models. When agents have quasi-linear preferences, his result can be applied to the auction model. 10 See Section 6 for the formal definition. 11 Recall that the payment of an agent under the VCG rule is decomposed into two parts. The first part is what is called Vickrey price, the social opportunity cost to allocate him an object; the second part is the term that is independent of his preference. Individual rationality and no subsidy for losers imply that the second part is zero. See also Chew and Serizawa (2007). 12 See also Vives (1987) and Hayashi (2013) for mathematical arguments. 13 See Saitoh and Serizawa (2008) for numerical examples.

4 448 Morimoto and Serizawa Theoretical Economics 10 (2015) license auctions, license prices often equal or exceed bidders annual revenues. Thus, bidders preferences are non-quasi-linear for such important auctions. 14 As contrasted with Holmström (1979), our result applies to such environments. Saitoh and Serizawa (2008) investigate a problem similar to ours in the case where the domain includes non-quasi-linear preferences and there are multiple copies of the same object. They generalize Vickrey rules by employing compensated valuations from no object and no payment, and characterize the generalized Vickrey rule by strategyproofness, efficiency, individual rationality, and no subsidy. 15 We stress that when preferences are not quasi-linear, the heterogeneity of objects makes the MPW rule different from the generalized Vickrey rule. 16 Although the assumption of quasi-linearity neglects the serious effects of large-scale auction payments in actual practice, it is difficult to investigate the above question without this assumption. Quasi-linearity simplifies the description of efficient allocations. More precisely, under quasi-linear preferences, an efficient allocation of objects can be achieved simply by maximizing the sum of realized benefits from objects (agents net benefits), and hence, is independent of how much agents pay. In this sense, Holmström (1979) characterizes only the payment part of strategy-proof and efficient rules. On the other hand, without quasi-linearity, efficient allocations of objects do depend on payments and thus, cannot be simply identified in the same way as in the quasi-linear case. In this article, we overcome that difficulty. Furthermore, as mentioned earlier, on non-quasi-linear domains, the MPW rule is different from the generalized Vickrey rule, and the former outperforms the latter in terms of our desirable properties, i.e., strategy-proofness and efficiency are satisfied by the MPW rule, but not by the generalized Vickrey rule. Needlessto say, Holmström s (1979) results cannot be applied to prove our results on the non-quasi-linear domain. It is worthwhile to mention that most standard results of auction theory, such as the revenue equivalence theorem, also depend on assuming quasi-linearity. Recently, Baisa (2013) studies an auction model where probabilistic allocations are accommodated and he demonstrates that the effect of non-quasilinearity makes optimal mechanisms qualitatively different. Since Hurwicz s (1972) seminal work, many authors have investigated efficient and strategy-proof rules in pure exchange economies. 17 In pure exchange economies, classical 18 preferences are standard, but no rule is strategy-proof, efficient, and individually rational on the classical domain. On the other hand, Demange and Gale (1985) show that, in the model studied in this article, the MPW rule is strategy-proof, efficient, 14 Ausubel and Milgrom (2002) also discuss the importance of the analysis under non-quasi-linear preferences. See Baisa (2013) for more examples of non-quasi-linear preferences. 15 Sakai (2008) also obtains a result similar to theirs. 16 In Section 6, we give a detailed discussion on this point by contrasting the MPW rule with the generalized Vickrey rule. 17 For example, see Zhou (1991), Barberà and Jackson (1995), Schummer (1997), Serizawa (2002), and Serizawa and Weymark (2003). 18 In pure exchange economies, where consumption spaces are some multidimensional Euclidean space, classical preferences are assumed to satisfy convexity in addition to continuity and monotonicity. Clearly, the class of such preferences contains non-quasi-linear preferences.

5 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 449 and individually rational on the classical domain. 19 Generalizing the MPW rule to the situations where price ranges are bounded, Andersson and Svensson (2014) introduce the minimum rationing price equilibrium rule, and demonstrate that it satisfies (group) strategy-proofness and a weak variant of efficiency. Miyake (1998) shows that only the MPW rule satisfies strategy-proofness among Walrasian rules. 20 Note that the Walrasian rules are a small part of the class of allocation rules satisfying efficiency, individual rationality, and no subsidy for losers. By developing analytical tools different from Miyake s (1998), 21 we extend his characterization in that we establish the uniqueness of the rules satisfying the desirable properties without confinement to Walrasian rules. Many authors have analyzed SA auctions in quasi-linear settings (e.g., Gul and Stacchetti 2000, Ausubel and Milgrom 2002, Ausubel 2004, 2006, de Vries et al. 2007, Mishra and Parkes 2007, Andersson et al. 2013). In non-quasi-linear settings, the MPW rules differ from the generalized Vickrey rules, and it is the MPWE allocation that coincides with the outcome of the SA auction. Alaei et al. (2013) construct an alternative algorithm computing MPWE in non-quasi-linear settings. Our result demonstrates that the SA auction and alternative algorithms analyzed by those authors are more important in non-quasi-linear settings. The problems of allocating objects and money have been studied by many authors. One of the extensively studied problems not referenced above is the one of fair (envyfree) allocation (Svensson 1983, Maskin 1987, Alkan et al. 1991, Tadenuma and Thomson 1991). 22 In the context of strategy-proofness, fair allocation rules are investigated by Tadenuma and Thomson (1995), Sun and Yang (2003), Ohseto (2006), and Svensson (2004, 2009). 23 When Svensson (2004, 2009) characterizes the class of strategy-proof and envy-free rules, he does not impose no subsidy for losers on rules, but imposes only the nonnegativity of the sum of payments the requirement that the sum of the agents payments be nonnegative. 24 This alternative requirement is mild and natural. However, we emphasize that envy-freeness is a strong requirement in his model and in ours. When each object is assigned to some agent, envy-freeness implies efficiency (Svensson 1983) and is almost equivalent to Walrasian equilibrium conditions. Given an allocation such that each object is assigned to some agent, take the price vector such that the price of each object is the payment of the agent who receives it. Envy-freeness implies that for this 19 More precisely, Demange and Gale (1985) study two-sided matching markets that contain our model as a special case and show that the rules selecting an optimal stable assignment for one side of the market are group strategy-proof for the agents on that side. 20 A Walrasian rule is the rule that assigns a Walrasian equilibrium allocation to each preference profile. 21 In Appendix B, we discuss why different analytical tools are necessary. 22 Envy-freeness (Foley 1967) is the requirement that no agent should prefer anyone else s assignment to his own. 23 Some authors also investigate the problem by other fairness axioms. See, for example, Ashlagi and Serizawa (2012) and Mukherjee (2014) for the axiom of anonymity in welfare, and see Sakai (2013) and Adachi (2014) for the axiom of weak envy-freeness for equals. 24 To be precise, he requires that the sum of the agents payments have a lower bound. This requirement implies that the total subsidy is limited by a prespecified level, but not that the subsidy to an individual agent is limited.

6 450 Morimoto and Serizawa Theoretical Economics 10 (2015) price vector, each agent demands the object he receives in the given allocation. Since we do not impose envy-freeness on rules, our results and the results of Svensson (2004, 2009) are logically independent. Other authors have investigated the existence of strategy-proof and nonbossy rules. 25 Miyagawa (2001) characterizes the class of strategy-proof, nonbossy, individually rational, and onto rules. Svensson and Larsson (2002) characterize the classes of strategyproof and nonbossy rules with several additional desirable properties. 26 It is well known that nonbossiness together with strategy-proofness makes the analysis tractable. Since the MPW rules violate nonbossiness, we do not impose this demanding property and thus, cannot apply their proof techniques in our proof. 1.4 Organization The article is organized as follows. Section 2 sets up the model and introduces basic concepts. Section 3 defines the MPWE and discusses its properties. Section 4 provides our main result. Section 5 defines the SA auction, and shows that its outcome coincides with the MPWE. Section 6 introduces the generalized Vickrey rules and contrasts them with the MPW rules. Section 7 concludes. Most proofs appear in the Appendix. Proofs omitted from the main paper are given in a supplementary file on the journal website, 2. The model and definitions There are n agents and m objects, where 2 n< and 1 m<. We denote the set of agents by N {1 n} and the set of objects by M {1 m}. LetL {0} M. Each agent consumes one object at most. We denote the object that agent i N receives by x i L. Object0 is referred to as the null object, and x i = 0 means that agent i receives no real object. We denote the amount that agent i pays by t i R. Foreachi N, agenti s consumption set is L R, anda(consumption) bundle for agent i is a pair z i (x i t i ) L R. Let0 (0 0). Each agent i has a complete and transitive preference relation R i on L R. LetP i and I i, respectively, be the strict relation and the indifference relation associated with R i. Given a preference R i and a bundle z i, let the upper contour set and lower contour set of R i at z i be UC(R i z i ) {z i L R : z i R i z i } and LC(R i z i ) {z i L R : z i R i z i }, respectively. For each i N,agenti s preference R i satisfies the following properties. Money monotonicity. For each x i L and each t i t i R, if t i < t i, then (x i t i )P i (x i t i ). Finiteness. For each t i R and each x i x i L, there exist t i t i (x i t i )R i (x i t i ) and (x i t i )R i (x i t i ). R such that 25 Nonbossiness (Satterthwaite and Sonnenschein 1981) is the requirement that when an agent s preferences change, if his assignment remains the same, then the chosen allocation should remain the same. 26 See also Schummer (2000) for the other analysis of strategy-proof and nonbossy rules.

7 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 451 Continuity. Foreachz i L R, UC(R i z i ) and LC(R i z i ) both are closed. Let R E denote the class of money monotonic, finite, and continuous preferences the extended domain. Given R i R E, z i (x i t i ) L R, andy i L, wedefinethe compensating valuation cv i (y i ; z i ) of y i from z i for R i by (y i t i + cv i (y i ; z i )) I i z i,andwe let CV i (y i ; z i ) t i + cv i (y i ; z i ). We refer to CV i (y i ; z i ) as the compensated valuation of y i from z i for R i. Note that by continuity and finiteness, CV i (y i ; z i ) exists, and by money monotonicity, CV i (y i ; z i ) is unique. The compensated valuation for R i is denoted by CV iẇe introduce another property of preferences. Desirability of objects. Foreachx i M and each t i R, (x i t i )P i (0 t i ). 27 Definition 1. A preference R i is classical if it satisfies money monotonicity, finiteness, continuity, and desirability of objects. Let R C denote the class of classical preferences the classical domain. Note that R C R E. Definition 2. A preference R i is quasi-linear if there is a valuation function v i : L R + such that (i) v i (0) = 0, (ii) for each x M, v i (x) > 0, and (iii) for each z i (x i t i ) L R and each z i (x i t i ) L R, z i R i z i if and only if v i(x i ) t i v i (x i ) t i. Let R Q denote the class of quasi-linear preferences the quasi-linear domain. Note that R Q R C. An object allocation is an n-tuple (x 1 x n ) L n such that for each i j N,ifx i 0 and i j, thenx i x j, that is, no two agents receive the same object except when both receive the null object. Let X be the set of object allocations. A (feasible) allocation is an n-tuple z (z 1 z n ) ((x 1 t 1 ) (x n t n )) [L R] n of bundles such that (x 1 x n ) X. LetZ be the set of feasible allocations. We denote the object allocation and the agents payments at z Z by x (x 1 x n ) and t (t 1 t n ), respectively. Let R be a class of preferences such that R R E.Apreference profile is an n-tuple R (R 1 R n ) R n.givenr (R 1 R n ) R n and N N, letr N (R i ) i N and R N (R i ) i N\N. An allocation rule, orsimplyarule, onr n is a function f from R n to Z. Givenarule f and a preference profile R R n, we denote agent i s assigned object under f at R by (R) and denote his payment by f t(r), and we write f x i i f i (R) (f x i (R) f t i (R)) f (R) (f 1(R) f n (R)) and f x (R) (f x j (R)) j N We introduce basic properties of rules. The efficiency condition defined below takes the auctioneer s preference into account and assumes that he is only interested in his revenue. An allocation z Z (Pareto-) dominates z Z for R R n if 27 A preference R i satisfies weak desirability of objects if for each x i M, (x i 0)P i 0. All the results in this article still hold if desirability of objects is replaced by weak desirability of objects.

8 452 Morimoto and Serizawa Theoretical Economics 10 (2015) (i) i N t i i N t i (ii) for each i N z i R i z i,and (iii) for some j N, z j P j z j. An allocation z Z is (Pareto) efficient for R R n if there is no feasible allocation that dominates z for R. Efficiency. ForeachR R n, f(r)is efficient for R. Individual rationality says that a rule should never select an allocation at which some agent is worse off than if he had received the null object and paid nothing. No subsidy says that the payments should always be nonnegative. No subsidy for losers says that the payments of agents who obtain the null object should always be nonnegative. No subsidy implies no subsidy for losers. Individual rationality. ForeachR R n and each i N, f i (R) R i 0. No subsidy. ForeachR R n and each i N, fi t (R) 0. No subsidy for losers. ForeachR R n and each i N,iff x i (R) = 0,thenft i (R) 0. The two properties below have to do with incentives. First, by misrepresenting his preferences, no agent should obtain an assignment that he prefers. Strategy-proofness. For each R R n, each i N, and each R i R, f i (R) R i f i (R i R i). The second property is stronger: by jointly misrepresenting their preferences, no group of agents should obtain assignments that they prefer. Group strategy-proofness. For each R R n and each N N, there is no R N R N such that for each i N, f i (R N R N )P i f i (R) Minimum price Walrasian equilibrium 3.1 Definition of Walrasian equilibria We define Walrasian equilibrium and minimum price Walrasian equilibrium. Let R R E in this section. All results in this section also hold on the classical domain R C. Let p (p 1 p m ) R m + be a price vector. The budget set at prices p is defined as B(p) {(x p x ) : x L}, wherep x = 0 if x = 0. Giveni N, R i R, andp R m +,agenti s demand set is defined as D(R i p) {x L : for each y L (x p x )R i (y p y )}. 28 Let A denote the cardinality of set A.

9 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 453 Definition 3. Let R R n.apair((x t) p) Z R m + is a Walrasian equilibrium for R if (WE-i) for each i N, x i D(R i p)and t i = p x i,and (WE-ii) for each y M, ifforeachi N, x i y, thenp y = 0. Condition (WE-i) says that each agent receives an object he demands and pays its price. Condition (WE-ii) says that an object s price is zero if it is not assigned. Fact 1. For each R R n, there is a Walrasian equilibrium for R. Fact 1 is already known. 29 Given R R n,letw(r)be the set of Walrasian equilibria for R,andletZ(R) and P(R) be the sets of Walrasian equilibrium allocations and prices for R, respectively, i.e., Z(R) {z Z : for some p R m + (z p) W(R)} and P(R) {p R m + : for some z Z (z p) W(R)} Next is a first welfare theorem for our model. 30 Fact 2. Let R R n and z Z(R). Thenz is efficient for R. 31 Fact 3 says that for each preference profile, there is a unique minimum Walrasian equilibrium price vector. The minimum price Walrasian equilibrium (hereafter MPWE) is the Walrasian equilibria associated with the minimum price. Fact 3(Demange and Gale 1985). For each R R n, there is a unique p P(R) such that for each p P(R), p p. Let p min (R) denote this price vector for R. Given R R n,letw min (R) be the set of minimum price Walrasian equilibria for R and let Z min (R) { z Z : (z p min (R)) W min (R) } 29 For example, see Alkan and Gale (1990). Our model is a special case of theirs. 30 See also Svensson (1983). 31 To see this, suppose that z (z 1 z n ) is not efficient for R. Then there is z (z 1 z n ) such that (i) i N t i i N t i (ii) for each i N, z i R i z i (iii) for some j N, z j P j z j. Since z Z(R), there is a price vector p R m + such that (z p) W(R). Then, by (ii) and (WE-i), for each i N, t i px i. By (iii) and (WE-i), t j <px j.thus, i N t i < i N px i = i N t i. This contradicts (i).

10 454 Morimoto and Serizawa Theoretical Economics 10 (2015) object B R 3 p B R 1 z 2 R 2 R 1 object A p A z 1 CV 3 (B; 0) null object 0 = z 3 CV 2 (A; 0) CV 2 (B; 0) CV 3 (A; 0) CV 1 (A; 0) CV 1 (B; 0) payment Figure 1. Illustration of non-quasi-linear preferences and the minimum price Walrasian equilibrium. By Facts 1 and 3,foreachR R n,thesetz min (R) is nonempty. Although the correspondence Z min is set-valued, it is essentially single-valued, i.e., for each R R n, each pair z z Z min (R), andeachi N, z i I i z i.32 As Demange et al. (1986), e.g., show for the quasi-linear domain, and as shown for our domain (Section 5), the SA auctions achieve the MPWE. 3.2 Illustration of minimum price Walrasian equilibrium Figure 1 illustrates an MPWE for three agents, and two objects, say A and B. Thereare three horizontal lines. The lowest one corresponds to the null object. The middle and highest lines correspond to the real objects A and B, respectively. The intersection of the vertical line and each horizontal line denotes the bundle consisting of the corresponding object and no payment. For example, the origin 0 denotes the bundle consisting of the null object and no payment. For each point z i on one of the three horizontal lines, the distance from z i to the vertical line denotes payment. For example, z 1 denotes the bundle consisting of object A and payment p A. Indifference between bundles is shown by a curvy line connecting them. Welfare increases with decreasing payments. Thus, in Figure 1, agent1prefersz 1 to 0. Assume that preferences are as depicted in Figure 1. The compensated valuations from the origin are ranked as CV 1 (A; 0) >CV 3 (A; 0) >CV 2 (A; 0) and CV 1 (B; 0) > 32 An allocation z Z is obtained by an indifferent permutation from z Z if there is a permutation π on N such that for each i N, z i = z π(i) and z i I i z i (Tadenuma and Thomson 1991). Note that for each pair z z Z min (R), z is obtained by an indifferent permutation from z.

11 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 455 CV 2 (B; 0) >CV 3 (B; 0). In Figure 1, agent 1 s preference is not quasi-linear, but classical. 33 Thus, Figure 1 also illustrates that R Q R C. The MPWE for the preference profile R = (R 1 R 2 R 3 ) is as follows: Agent 1 receives object A and pays CV 3 (A; 0), i.e., the price p A of object A is CV 3 (A; 0). His consumption is z 1. Agent 2 receives object B and pays CV 1 (B; z 1 ), i.e., the price p B of object B is CV 1 (B; z 1 ). His consumption is z 2. Agent 3 s consumption is 0 and is depicted as z 3. Let us see why the allocation z (z 1 z 2 z 3 ) is an MPWE for R. First, note that for each agent i = 1 2 3, z i is maximal for R i in the budget set {0 (A p A ) (B p B )}. Thus,z is a Walrasian equilibrium. Next, let (p A p B ) be a Walrasian equilibrium price vector. We show p A p A and p B p B.Ifp A <p A and p B <p B, then all agents prefer (A p A ) or (B p B ) to 0, that is, all three agents demand A or B or both. In that case, one agent cannot receive an object he demands, contradicting (WE-i) in Definition 3. Thus,p A p A or p B p B.If p A <p A,thenp B p B, and so both agents 1 and 3 prefer (A p A ) to 0 and (B p B ), that is, both demand only A. In that case, agents 1 or 3 cannot receive the object they demand, contradicting Walrasian equilibrium. Therefore, p A p A. Ifp B <p B, both agents1and2prefer(b p B ) to 0 and (A p A ), and so agents 1 or 2 cannot receive the object they demand, contradicting Walrasian equilibrium. Therefore, p B p B.Hence, (z p) is the MPWE. 3.3 Overdemanded and underdemanded sets Next, we introduce the concepts of overdemanded set and underdemanded set (Mishra and Talman 2010, e.g.), and relate these concepts to Walrasian equilibria. Definition 4. (i) A set M M of objects is (weakly) overdemanded at p for R if {i N : D(Ri p) M } ( )> M (ii) A set M M of objects is (weakly) underdemanded at p for R if [ x M p x > 0] {i N : D(R i p) M } ( )< M In Figure 1, note that {i N : D(R i p) {A}} =, {i N : D(R i p) {B}} = {2}, {i N : D(R i p) {A B}} = {1 2}, {i N : D(R i p) {A} }={1 3}, {i N : D(R i p) {B} } ={1 2}, and{i N : D(R i p) {A B} } ={1 2 3}. Thus, no set is overdemanded or weakly underdemanded. Fact 4 and Theorem 1 below are established by Mishra and Talman (2010) forquasilinear preferences. Fact 4 is a characterization of Walrasian equilibria by means of the concepts of overdemanded and underdemanded sets. Their proof also works for Fact 4 in the extended domain. 33 Suppose that agent 1 s preference is quasi-linear. Then since CV 1 (B 0) >CV 1 (A 0), agent1 scompensated valuation CV 1 (B z 1 ) of object B from the point z 1 in Figure 1 must be greater than CV 2 (B 0). However, in Figure 1, agent1prefersz 1 to the point (B CV 2 (B 0)). This is a contradiction.

12 456 Morimoto and Serizawa Theoretical Economics 10 (2015) Fact 4(Mishra and Talman 2010). Let R R n.apricevectorp is a Walrasian equilibrium price vector for R if and only if no set is overdemanded and no set is underdemanded at p for R. Theorem 1 is a characterization of the minimum price Walrasian equilibrium by means of the concepts of overdemanded and weakly underdemanded sets. We emphasize, in contrast to Fact 4, thatmishra and Talman s (2010) proof crucially depends on quasi-linearity. It relies on the simple fact that when preferences are quasi-linear, if a set M is weakly underdemanded at a Walrasian equilibrium price vector p, then all the prices of M can be slightly lowered by the same amount while maintaining the Walrasian equilibrium conditions (WE-i) and (WE-ii). 34 However, this is not true when preferences are not quasi-linear. Theorem 1 is a novel result, and is the key to obtaining Theorem 2 and Proposition 1. Theorem Let R R n.apricevectorp is a minimum Walrasian equilibrium price vector for R if and only if no set is overdemanded and no set is weakly underdemanded at p for R. Corollary 1 says that if the number of objects is greater than or equal to the number of agents, the price of some objects is 0. ItisusedtoproveFact 6. Corollary 2 says that each object whose price is positive is connected by agents demands to the null object or to an object with a price of 0. This corollary is used to prove Theorem For example, in Figure 1, object B has a positive equilibrium price, agent 1 s demand connects objects A and B, and agent 3 s demand connects object A and the null object. Corollary 1 (Existence of free object). Let m n, R R n,andz Z min (R). Then there is i N such that p x i min (R) = 0. Corollary 2 (Demand connectedness). 37 Let R R n and (z p) W min (R). Foreach x M with p x > 0, there is a sequence {i k } K k=1 of K distinct agents such that (i) x i 1 = 0 or p x i 1 = 0, (ii) for each k {2 K 1}, x ik 0 and p x i k > 0, (iii) x ik = x, and (iv) for each k {1 K 1}, {x ik x ik+1 } D(R ik p). Here, we also introduce a concept of d i -truncation of a preference. This concept is important to prove Theorem 1. It says that the welfare position of each bundle z i M R is lowered as much as d i in terms of money, but their relative positions are kept. Given R i R and d i R, thed i -truncation of R i is the preference R i such that for each z i M R, CV i (0; z i) = CV i (0; z i ) + d i.givenr R n and d R n,thed-truncation of R is the preference profile R such that for each i N, R i is the d i-truncation of R i. The following remark and fact pertain to truncations. Remark 1(i) and Fact 5 are used to prove Theorem For details, refer to the proof of Lemma 3 in Mishra and Talman (2010). 35 Alaei et al. (2013) also establish this result independently by using different proof methods. 36 See Lemma 12 for details. 37 This structure is discussed by Demange et al. (1986) and Miyake (1998).

13 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 457 Remark 1. Let R i R, d i R, andr i be the d i-truncation of R i. Then the following statements hold: (i) For each z i ẑ i M R, z i R i ẑ i if and only if z i R i ẑi. (ii) R i satisfies money monotonicity, finiteness, and continuity, and so R i RE. (iii) For large d i, R i violates desirability of objects.38 Fact 5(Roth and Sotomayor 1990). Let R R n and let R be a d-truncation of R such that for each i N, d i 0. Thenp min (R ) p min (R). 4. Main results In this section, we provide a characterization of the MPWE by means of properties of rules. Let R R E. Definition 5. A rule f on R n is a minimum price Walrasian (MPW) rule if for each R R n, f(r) Z min (R). 4.1 Properties of the minimum price Walrasian rule Let g be an MPW rule on R n. First, by Fact 2, foreachr R n, g(r) is efficient for R. Let R R n. Then there is a price vector p (p 1 p m ) R m + such that for each i N, (a) g i (R) B(p), and(b)foreachz i B(p), g i(r) R i z i. Let i N. Note that, for each x M, p x 0 and B(p) ={(0 0) (1 p 1 ) (2 p 2 ) (m p m )}. Thus, by (a), gi t (R) 0, and by (b), g i (R) R i 0. Therefore, the MPW rule satisfies efficiency, individual rationality, and no subsidy. Fact 6(Demange and Gale 1985). The minimum price Walrasian rule is group strategyproof. Theorem 1 allows a direct proof (see Appendix B). 4.2 Characterizations In this subsection, we assume that each agent has a classical preference and the number of agents exceeds the number of objects. Recall that all results established in Section 3 also hold in this case. Theorem 2 is our main result of this article, a characterization of the MPW rule. Theorem 2. Let R R C and n>m. A rule f on R n satisfies strategy-proofness, efficiency, individual rationality, and no subsidy for losers if and only if it is a minimum price Walrasian rule: for each R R n, f(r) Z min (R). 38 Because of Remark 1(iii), a d i -truncation of a classical preference may not be classical. However, this does not create any problems in the proofs of this article.

14 458 Morimoto and Serizawa Theoretical Economics 10 (2015) The proof is given in Appendix B. Since the MPW rules are group strategy-proof, Theorem 2 implies that only the MPW rules satisfy group strategy-proofness, efficiency, individual rationality, and no subsidy for losers. Since no subsidy implies no subsidy for losers, Theorem 2 also implies that only the MPW rules satisfy strategy-proofness, efficiency, individual rationality, and no subsidy. 4.3 Indispensability of the axioms and assumptions The only if part of Theorem 2 fails if we drop any of the four axioms, as shown by the following examples. Example 1 (Dropping strategy-proofness). Let f betherulethatchoosesa maximum price Walrasian equilibrium allocation for each preference profile. Then f satisfies the axioms of Theorem 2 except for strategy-proofness. 39 Example 2 (Dropping efficiency). Let f betherulesuchthatforeachpreferenceprofile, each agent receives the null object and pays nothing. Then f satisfies the axioms of Theorem 2 except for efficiency. Next, we introduce variants of Walrasian equilibria those with entry fees. Given an entry fee e i R, letd(r i p e i ) {x L : for each y L (x p x + e i )R i (y p y + e i )}, where p x = 0 if x = 0. Apair((x t) p) Z R m is a Walrasian equilibrium with entry fees for R R n if there is an entry fee vector e = (e 1 e n ) R n such that (WE-i*) for each i N x i D(R i p e i ),andt i = p x i + e i,and (WE-ii) for each y M, ifforeachi N, x i y,thenp y = 0. Note that, similarly to Facts 1, 2, and3, for each preference profile R R n and each e = (e 1 e n ) R n, there is an MPWE with entry fees e, and it is efficient. A rule f is a minimum price Walrasian rule with entry fees if there is an entry fee vector e R n and for each R, f(r) is an MPWE with entry fees e. Then, MPW rules with entry fees are efficient. Similarly to Fact 6, we can show that they are also group strategy-proof. Example 3 (Dropping individual rationality). Let e = (e 1 e n ) R n be an entry fee vector such that for each i N, e i > 0. Then the associated minimum price Walrasian rule with entry fees satisfies the axioms of Theorem 2 except for individual rationality. Example 4 (Dropping no subsidy for losers). Let e = (e 1 e n ) R n be an entry fee vector such that for each i N, e i < 0. Then the associated minimum price Walrasian rule with entry fees satisfies the axioms of Theorem 2 except for no subsidy for losers. 39 Demange and Gale (1985) also show that for each preference profile, there is a maximum price Walrasian equilibrium. When there is only one object, the maximum price Walrasian equilibrium corresponds to the first price auction. It is well known that the first price auction is not strategy-proof.

15 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 459 We further generalize MPW rules with entry fees as follows: a rule f is a minimum price Walrasian rule with variable entry fees if there is a list {e i ( )} i N of entry fee functions defined on R n, and for each R, f(r)is an MPWE with entry fees {e i (R)} i N. MPW rules with variable entry fees are also efficient. Note that if for each i N, theentryfeefunctione i ( ) depends only on the other agents preferences R i, then the associated MPW rule with variable entry fees {e i ( )} i N on the quasi-linear domain is strategy-proof and so, by Holmström (1979), it is a rule, called Groves rule. 40 However, as illustrated in Example 5, an MPW rule with variable entry fees {e i ( )} i N is not strategyproof on the classical domain even if for each i N, the entry fee function e i ( ) depends only on the other agents preferences R i. This fact demonstrates the complexity of analysis on the classical domain. Example 5 (A violation of strategy-proofness of an MPW rule with variable entry fees). Let N {1 2} and M {1}. Let f be the MPW rule with variable entry fees {e i ( )} i N such that for each R 2, e 1 (R 2 ) = 0, and for each R 1, e 2 (R 1 ) = CV 1 (1; 0). LetR be a preference profile such that CV 1 (1; 0) 4, cv 2 (1; (0 4)) 2, andcv 2 (1; (0 7)) 1. Then f 1 (R) = (1 2). Let R 1 be such that CV 1 (1; 0) 7. Then f 1(R 1 R 1) = (1 1). Thus, f 1 (R 1 R 1)P 1 f 1 (R). One might wonder if the MPW rules with entry fees can be characterized by only strategy-proofness and efficiency. Our proof of Theorem 2 fails if individual rationality and no subsidy for losers are dropped. However, we have not found an example of a rule that satisfies strategy-proofness and efficiency, but is not an MPW rule with entry fees. Therefore, it is an open question whether the class of MPW rules with entry fees can be characterized by only strategy-proofness and efficiency. One might also wonder if the assumption that n>mcan be dropped in Theorem 2. Our proof of Theorem 2 also fails if n m. However, we have not found an example of a rule that satisfies the four axioms of Theorem 2, but is not an MPW rule even if n>mis dropped. Therefore, this question is also open. 5. Simultaneous ascending auction We define a class of simultaneous ascending auctions and show that they achieve the MPWE. Let R R E, R R n,andp R m +. Definition 6. A set M M is a minimal overdemanded set at p for R if M is overdemanded at p for R and there is no M M such that M is overdemanded at p. Under a (continuous time) simultaneous ascending auction, there is a constant d> 0, and at each time, each bidder submits his demand at the current price vector and the prices of the objects in a minimal overdemanded set are raised at a speed at least d. When there is no overdemanded set, the auction stops. Given a preference profile, a simultaneous ascending auction generates a price path. 40 See Section 6 for the definition of Groves rule.

16 460 Morimoto and Serizawa Theoretical Economics 10 (2015) Definition 7. A simultaneous ascending (SA) auction is a function τ from R + R m + R n to R m + such that the following statements hold: (i) Given R R n, τ is integrable with respect to time t R + and price p R m +. (ii) There is d>0 such that for each t R +,eachp R m +,eachr Rn,andeach x M, (ii-a) if x is in a minimal overdemanded set at p,thenτ x (t p R) d (ii-b) τ x (t p R)= 0 otherwise. For each R R n,theprice path generated by an SA auction τ is a function p from R + to R m + such that the following statements hold: (i) For each x M, p x (0) = 0. (ii) For each x M and each t R +, p x (t) = t 0 τ x (s p(s) R) ds Proposition 1 says that the outcome of an SA auction coincides with the MPWE. Proposition 1. For each R R n, the price path generated by any simultaneous ascending auction converges to the minimum Walrasian equilibrium price in a finite time. The proof is given in Appendix C. Proposition 1 implies that for each R R n,the price path p( ) generated by an SA auction has a termination time T such that for each t T, p(t) = p(t ) = p min (R), and at the final prices p(t ), each agent receives an object from his demand and pays the final price of the object that he receives. 6. Generalized Vickrey rule In this section, we introduce the generalized Vickrey rules and contrast them with the MPW rules. 6.1 Generalized Vickrey rule Each quasi-linear preference R i can be defined by means of a valuation function v i : L R +, and a preference profile R in the quasi-linear domain corresponds to a valuation profile v(r) (v 1 (R 1 ) v n (R n )). Given a valuation profile v = (v 1 v n ),let (x 1 (v) x n (v)) arg max (x 1 x n ) X max (x1 x n ) X as follows. i v i(x i ), σ i (v) j i v j(x j (v)) and σ i (v) j i v j(x j ). 41 On the quasi-linear domain, the Vickrey rules are defined 41 Note that effectively σ i ( ) is a function of v i.

17 Theoretical Economics 10 (2015) Strategy-proofness and efficiency 461 Definition 8. A rule f on the quasi-linear domain is a Groves rule if (i) for each valuation profile v, f x (v) arg max (x1 x n ) X i v i(x i ), and (ii) for each i N, thereisa function h i of the other agents valuation profile v i such that for each valuation profile v, fi t(v) = h i(v i ) σ i (v). AGrovesrulef is a Vickrey rule if for each i N, h i = σ i. To generalize Vickrey rules to the classical domain, we need to use some valuation function v i for each classical preference R i. The compensated valuation CV i ( ; 0) from the origin is defined for each classical preference R i and a generalization of valuation function, and so is a natural candidate. Given a classical preference R i,letv i ( ; R i ) be a function defined as, for each x L, v i (x; R i ) CV i (x; 0). Given a classical preference profile R,letv (R) (v 1 ( ; R 1 ) v n ( ; R n )). Definition 9. A rule f on the classical domain is a generalized Vickrey rule if for each classical preference profile R, f x (v (R)) arg max (x1 x n ) X i v i(x i ; R i ), and for each i N, fi t(v (R)) = σ i (v (R)) σ i (v (R)). A classical preference R i is object-blind if for each x y M and each t R, (x t) I i (y t). We call the class of object-blind preferences the object-blind domain. The object-blind domain is a subset of the classical domain. On the object-blind domain, Saitoh and Serizawa (2008) andsakai (2008) characterize the generalized Vickrey rules. Fact 7(Saitoh and Serizawa 2008, Sakai 2008). Let n>m. A rule on the object-blind domain satisfies strategy-proofness, efficiency, individual rationality, and no subsidy if and only if it is a generalized Vickrey rule. 42 On the quasi-linear domain, the classes of Vickrey rules, generalized Vickrey rules, and MPW rules coincide. Fact 7 suggests that the generalized Vickrey rules are natural generalizations of the Vickrey rules on the object-blind domain. On the object-blind domain, the classes of generalized Vickrey rules and MPW rules also coincide. However, these two classes of rules differ outside the above two domains, as explained in Section 6.2. Thus, Fact 7 does not imply Theorem 2. Since the object-blind domain is smaller than the classical domain, Theorem 2 does not imply Fact 7 either. Therefore, the two results are mathematically independent. 6.2 Generalized Vickrey rule vs. minimum price Walrasian rule Notice that, in example of Section 3.2 (Figure 1), agent 2 s payment in the MPWE allocation z cannot be computed from the compensated valuations v i ( ; R i ), i = 1 2 3, from the origin 0. Payments of the MPW rule depend on the compensated valuations from various points. It is worthwhile to mention that for the preference profile in Figure 1, it is agent 1 s preference R 1 that determines whether agent 2 or agent 3 receives a real object in the MPWE allocation. In Figure 1, agent1prefers(a CV 3 (A; 0)) 42 It is straightforward that on the object-blind domain, strategy-proofness, efficiency, individual rationality, and no subsidy for losers imply no subsidy.

18 462 Morimoto and Serizawa Theoretical Economics 10 (2015) to (B CV 2 (B; 0)), and agent 2 receives a real object. However, if agent 1 prefers (B CV 2 (B; 0)) to (A CV 3 (A; 0)), agent 3 instead receives a real object. Object allocations of the MPW rule also depend on the compensated valuations from various points. Thus, the MPWE allocation z is not the outcome of the generalized Vickrey rule. Accordingly, the MPW rule does not coincide with the generalized Vickrey rule. 43 One can easily check that the generalized Vickrey rule is neither efficient nor strategy-proof on the classical domain with heterogeneous objects. To check this fact, let R 1 R C, R 2 R Q,andR 3 R Q be such that CV 1 (A; 0) = 9, CV 1 (B; 0) = 10, (A 6)P 1 (B 5), CV 2 (A; 0) = 3, CV 2 (B; 0) = 5, CV 3 (A; 0) = 6, andcv 3 (B; 0) = 2. The indiffence curves of Figure 1 illustrate those preferences. The outcome of the generalized Vickrey rule for R is z ((B 5) (0 0) (A 4)). Letz ((A 6) (B 5) (0 2)). Then z Pareto-dominates z, a violation of efficiency. Let R 1 RQ be such that CV 1 (A; 0) = 8 and CV 1 (B; 0) = 5. Then under the generalized Vickrey rule, the bundle that agent 1 obtains for (R 1 R 1) is (A 6). Since (A 6)P 1 (B 5), the generalized Vickrey rule violates strategy-proofness. The generalized Vickrey rule employs only a small part of the information about agents preferences (i.e., compensated valuations from the origin). On the other hand, the MPW rule employs other information (i.e., compensated valuations from various points). As we stated in Section 4, only the MPW rule satisfies strategy-proofness, efficiency, individual rationality, and no subsidy for losers on the domain including nonquasi-linear preferences. Thus, the information about compensated valuations from various points is necessary to design rules satisfying the above four properties on this domain. Proposition 1 states that the SA auction achieves the same outcome as the MPW rule. 7. Concluding remarks In this article, we mainly focus on the analysis of rules that allocate objects efficiently, and we show that only the MPW rules are desirable based on the four properties strategyproofness, efficiency, individual rationality, and no subsidy for losers. It would be also important to investigate rules that produce more revenues for the auctioneer. An interesting question relating to this issue is whether there are strategy-proof, efficient, and individually rational rules that produce greater revenues than the MPW rule for each preference profile. We hope that the results and techniques developed in this article will be useful for the study of this research topic. Appendixes: Proofs In this appendix, we provide the proofs of all results in the article. In Appendix A, we prove Theorem 1 and Corollaries 1 and 2. InAppendix B, wegivetheproofsofthemain results (Fact 6 and Theorem 2). Appendix C gives the proof of Proposition 1. The proofs of Facts 4 and 5 appear in the supplementary file on the journal website. 43 When agent 1 prefers (B CV 2 (B; 0)) to (A CV 3 (A; 0)), the MPWE allocation z (z 1 z 2 z 3 ) is z ((B CV 2 (B; 0)) (0 0) (A CV 1 (A; z 1 ))). Thus, unless CV 1(A; z 1 ) = CV 2(B; 0) + CV 1 (A; 0) CV 1 (B; 0), the MPWE allocation z does not coincide with the outcome of the generalized Vickrey rule.